#8106057

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#40517	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(3,7)(6,8)(9,10)(11,19)(12,23)(13,24)(14,20)(15,17)(16,22)(18,21)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,17,19)(12,18,24,20)(13,21,23,14)(26,30)(31,33)(32,37)(38,39);(2,5)(6,8)(10,17)(12,24)(16,19)(18,20)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,20)(14,16)(15,24)(17,23)(18,22)(19,21)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13)(10,12)(11,21)(14,22)(15,23)(16,18)(17,24)(19,20);(1,3)(2,6)(4,7)(5,8)(9,14,15,21)(10,18,17,20)(11,13,22,23)(12,16,24,19)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013253);